Different types of materials are used in many applications, such as noise reduction, thermal insulation, filtration, etc. For example, fibrous materials are often used in noise control problems for the purpose of attenuating the propagation of sound waves. Fibrous materials may be made of various types of fibers, including natural fibers, e.g., cotton and mineral wool, and artificial fibers, e.g., glass fibers and polymeric fibers such as polypropylene, polyester and polyethylene fibers. The acoustical properties of many types of materials are based on macroscopic properties of the bulk materials, such as flow resistivity, tortuosity, porosity, bulk density, bulk modulus of elasticity, etc. Such macroscopic properties are, in turn, controlled by manufacturing controllable parameters, such as, the density, orientation, and structure of the material. For example, macroscopic properties for fibrous materials are controlled by the shape, diameter, density, orientation and structure of fibers in the fibrous materials. Such fibrous materials may contain only a single fiber component or a mixture of several fiber components having different physical properties. In addition to the solid phase of the fiber components of the fibrous materials, a fibrous material's volume is saturated by fluid, e.g., air. Thus, fibrous materials are characterized as a type of porous material.
Various acoustical models are available for various materials, including acoustical models for use in the design of porous materials. Existing acoustical models for porous materials can generally be divided into two categories: rigid frame models and elastic frame models. The rigid models can be applied to porous materials having rigid frames, such as porous rock and steel wool. In a rigid porous material, the solid phase of the material does not move with the fluid phase, and only one longitudinal wave can propagate through the fluid phase within the porous materials. Rigid porous materials are typically modeled as an equivalent fluid which has complex bulk density and complex bulk modulus of elasticity. On the other hand, the elastic models can be applied to porous materials whose frame bulk modulus is comparable to that of the fluid within the porous materials, e.g., polyurethane foam, polyimide foam, etc. There are three types of waves that can propagate in an elastic porous material, i.e., two compressional waves and one rotational wave. The motions of the solid phase and the fluid phase of an elastic porous material are coupled through viscosity and inertia, and the solid phase experiences shear stresses induced by incident sound hitting the surface of the material at oblique incidence.
However, such rigid and elastic material models, some of which are described below, do not provide adequate modeling of limp fibrous materials, e.g., limp polymeric fibrous materials such as those comprised of, for example, polypropylene fibers and polyester fibers. The term "limp" as used herein refers to porous materials whose bulk elasticity, in vacuo, of the material is less than that of air.
The acoustical study of porous materials can be found as early as in Lord Rayleigh's study of sound propagation through a hard wall having parallel cylindrical capillary pores as described in Strutt et al., Theory of Sound, Vol. II, Article 351, 2.sup.nd Edition, Dover Publications, NY (1945). Models based on the assumption that the frame of the porous material does not move with the fluid phase of the porous material are categorized as the rigid frame porous models. Various rigid porous material models have been proposed, including those described in Monna, A. F., "Absorption of Sound by Porous Wall," Physica 5, pp. 129-142 (1938); Morse, P. M, and Bolt, R. H., "Sound Waves in Rooms," Reviews of Modern Physics 16, pp. 69-150 (1944); and Zwikker, C. and Kosten, C. W., Sound Absorbing Materials, Elsevier, N.Y. (1949). These models assumed, similar to Rayleigh's work, that the sound wave propagation within a rigid porous material can be described by using equations of motion and continuity of the interstitial fluid.
Rigid porous materials have also been modeled as an equivalent fluid having complex density, as described in Crandall, I. B., Theory of Vibrating Systems and Sound, Appendix A, Van Nostrand Company, NY (1927), and having complex propagation constants when viscous and thermal effects were considered. In Delany, M. E. and Bazley, E. N., Acoustical Characteristics of Fibrous Absorbent Materials," National Physical Laboratories, Aerodynamics Division Report, AC 37 (1969) the acoustical properties of rigid fibrous materials were studied differently. As described therein, a semi-empirical model of characteristic impedance and propagation coefficient as a function of frequency divided by flow resistance was established. This model was based on the measured characteristic impedance of fibrous materials having a wide range of flow resistance. In Smith, P. G. and Greenkorn, R. A., "Theory of Acoustical Wave Propagation in Porous Media," Journal of the Acoustical Society of America, Vol. 52, pp. 247-253 (1972), the effects of porosity, permeability (inverse of flow resistivity), shape factor and other macrostructural parameters on acoustical wave propagation in rigid porous media were investigated. Further, some rigid porous material theories applied the concept of complex density while others used flow resistance. A comparison of these two approaches was described in Attenborough, K., "Acoustical Characteristics of Porous Materials," Physics Reports, 82(3), pp. 179-227 (1982). In summary, the rigid porous material models only allow one longitudinal wave to propagate through the rigid medium and the rigid frame is not excited by the fluid phase within the porous material. Such rigid porous material models do not adequately predict the acoustical properties of limp porous materials.
As opposed to rigid porous models, elastic models of porous materials have also been described. By considering the vibration of the solid phase of a porous material due to its finite stiffness, Zwikker and Kosten arrived at an elastic model taking into account the coupling effects between the solid and fluid phases as described in Zwikker, C. and Kosten, C. W., Sound Absorbing Materials, Elsevier, N.Y. (1949). This work was extended by Kosten and Janssen, as described in Kosten, C. W. and Janssen, J. H., "Acoustical Properties of Flexible Porous Materials," Acoustica 7, pp. 372-378 (1957), which adapted the expression of complex density given by Crandall (1927) and complex density of air within pores given by Zwikker and Kosten (1949). A model that corrected the error of fluid compression effects in the work of Zwikker and Kosten (1949) and considered the oscillation of solid phase excited by normally incident sound has also been set forth. In this model, a fourth order wave equation indicated that two longitudinal waves can propagate in elastic porous materials as opposed to the single wave in rigid materials. In Shiau, N. M., "Multi-Dimensional Wave Propagation In Elastic Porous Materials With Applications To sound Absorption, Transmission and Impedance Measurement," Ph.D. Thesis, School of Mechanical Engineering, Purdue University (1991), Bolton, J. S., Shiau, N. M., and Kang, Y. J., "Sound Transmission Through Multi-Panel Structures Lined With Elastic Porous Materials," Journal of Sound and Vibration 191, pp. 317-347 (1996), and Allard, J. F., Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Elsevier Science Publishers Ltd., NY (1993) Biot's theory as described in Biot, M. A., "General Solutions of the Equations of Elasticity and Consolidation for a Porous Material," Journal of Applied Mechanics 23, pp. 91-96 (1956A); Biot, M. A., "Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low Frequency Range. II. Higher Frequency Range," Journal of the Acoustical Society of America 28, pp. 168-191 (1956B); and Biot, M. A., "The Elastic Coefficients of the Theory of Consolidation," Journal of the Applied Mechanics 24, pp. 594-601 (1957) in the field of geophysics, was adapted to develop elastic porous material models which allow the shear wave propagation through the elastic frame induced by obliquely incident sound to be considered. In these elastic models, the stress-strain relations and equations of motions of solid and fluid phases yield one fourth order equation governing two compressional waves and one second order equation governing one rotational wave.
However, when an acoustical wave propagates in a limp porous material, the vibration of the solid phase is excited only by the viscous and inertial forces through the coupling with the fluid phase. Due to lack of frame stiffness in such limp porous materials, no independent wave can propagate through the solid phase of the limp media. This fact leads to numerical singularities when the bulk stiffness in an elastic model is made small or set equal to zero in an attempt to model a limp porous material. Therefore, the types of waves in a limp material are reduced to only one compressional wave and the elastic models for limp porous materials are not adequate for use in design of limp porous materials.
Limp porous materials have been studied explicitly by a relatively small number of investigators; e.g., Beranek, L. L., "Acoustical Properties of Homogeneous, Isotropic Rigid Tiles and Flexible Blankets," Journal of the Acoustical Society of America 19, pp. 556-568 (1947), Ingard, K. U., "Locally and Nonlocally Reacting Flexible Porous Layers: A Comparison of Acoustical Properties," Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 103, pp. 302-313 (1985), and Goransson, P., "A Weighted Residual Formulation of the Acoustic Wave Propagation Through Flexible Porous Material and a Comparison with a Limp Material Model," Journal of Sound and Vibration 182, pp. 479-494 (1995).
There have also been attempts to develop acoustical models for fibrous materials: e.g., parallel resiliently supported fibers in Kawasima, Y., "Sound Propagation In a Fibre Block as a Composite Medium," Acustica, 10, pp. 208-217 (1960), and transversely stacked elastic fibers in Sides, D. J., Attenborough, K., and Mulholland, K. A., "Application of a Generalized Acoustic Propagation Theory to Fibrous Absorbents," Journal of Sound and Vibration, 19, pp. 49-64 (1971). The model of Kawasima (1960) results in a set of equations similar to those of Zwikker and Kosten (1949) and thus describe one-dimensional wave propagation in an elastic porous medium. By following this approach, limp materials can only be treated as a special case by setting the elasticity constants equal to zero, which may lead to numerical singularities. The model of Sides, Attenborough and Mulholland (1971) incorporates the Biot (1956B) model, but in a one-dimensional form, and it is assumed that the bulk solid phase has a finite stiffness. Thus, in this model properties of two longitudinal waves within the porous material are governed by a fourth order equation. Again numerical singularities would result if the bulk stiffness of the material were set equal to zero, i.e., if the material were assumed to be limp.
The macroscopic property, flow resistance, used in many of the models as described in the above cited and incorporated references, is one of the more significant properties of fibrous porous materials in determining their acoustical behavior. Therefore, the determination of flow resistance is of significant importance. In Nichols, R. H. Jr., Flow-Resistance Characteristics of Fibrous Acoustical Materials," Journal of the Acoustical Society of America, Vol. 19, No. 5, pp. 866-871 (1947), an expression of flow resistance in power law of fiber radius, material thickness and surface density is expressed. The power was determined experimentally and the value varied for different types of construction of the material. Delany and Bazley, in Delany, M. E. and Bazley, E. N., "Acoustical Characteristics of Fibrous Absorbent Materials," National Physical Laboratories, Aerodynamics Division Report, AC 37 (1969) and Delany, M. E. and Bazley, E. N., "Acoustical Properties of Fibrous Absorbent Materials," Applied Acoustics, Vol. 3, pp. 105-116 (1970), used measured flow resistance to establish a semi-empirical model for predicting the characteristic impedance of fibrous materials. Others, such as described in Bies, A. and Hansen, C. H., "Flow Resistance Information For Acoustical Design," Applied Acoustics, Vol. 13, pp. 357-391 (1980); Dunn, P. I. and Davern, W. A., "Calculation of Acoustic Impedance of Multi-Layer absorbers," Applied Acoustics, Vol. 19, pp. 321-334 (1986); and Voronia, N., "Acoustic Properties of Fibrous Materials," Applied Acoustics, Vol. 42, pp. 165-174 (1994) have also tried to predict the acoustic impedance of porous materials with empirical relations expressed purely in terms of flow resistance. In Ingard, K. U. and Dear, T. A., "Measurement of Acoustic Flow Resistance," Journal of Sound and Vibration, Vol. 103, No. 4, pp. 567-572 (1985), a method was proposed to measure the dynamic flow resistance of materials. It was found that the measured dynamic flow resistance is very close to the steady flow resistance at sufficiently low frequency. Woodcock and Hodgson, in Woodcock, R. and Hodgson, M., "Acoustic Methods For Determining the Effective Flow Resistivity of Fibrous Materials," Journal of Sound and Vibration, Vol. 153, No. 1, Feb. 22, pp. 186-191(1992) predicted the flow resistance by measuring the acoustic impedance. Besides the studies of the flow resistance and modeling thereof in the acoustical literature, there are other flow resistance studies in the fields of geophysics, aerosol science, and filtration.
Well known Darcy's law, as shown in Equation 1, gives the relation between the flow rate (Q) and pressure difference (.DELTA.p) defining flow resistance (W) for fibrous porous materials. In other words, flow resistance of a layer of fibrous porous material is defined as the ratio between the pressure drop (.DELTA.p) across the layer and the average velocity, i.e., steady flow rate (Q) through the layer. ##EQU1##
Therefore, flow resistivity (.sigma.) can be defined as shown Equation 2. ##EQU2##
wherein the variables shown therein and others included in flow resistivity Equations below are:
.DELTA.p, the pressure drop across the layer of material PA1 Q, the flow rate PA1 A, the area of the layer of material PA1 h, the thickness of the layer of material PA1 .eta., the viscosity of the gas PA1 .rho., the material's density PA1 .lambda., the mean free path of the material's molecules PA1 r, the mean radius of fibers of the material PA1 c, the packing density or solidity of the material.
Based on Darcy's law, Davies as described in Davies, C. N., "The Separation of Airborne Dust and Particles," Proc. Inst. Mech. Eng. 1B (5), pp. 185-213 (1952), derived the following functional relationship of Equation 3. ##EQU3##
The first term of the function expresses Darcy's law, the second term of the expression is referred to as Reynold's number, the third term is the packing density or solidity, and the fourth term is referred to as Knudsen's number. For fibrous materials, Knudsen's number and Reynold's number, are typically neglected. Therefore, Equation 4 results. ##EQU4##
From Equation 4, flow resistivity is as defined in Equation 5. ##EQU5##
Based on Equation 5, as described in Davies (1952), an empirical expression for flow resistivity is set forth as noted in Equation 6. ##EQU6##
Various other empirical relations have been expressed for flow resistivity. For example, in Bies and Hanson (1980), flow resistivity has been defined as shown in Equation 7. ##EQU7##
In addition, various other theoretical expressions for flow resistivity have been described. For example, in Langmuir, I., "Report on Smokes and Filters," Section I. U.S. Office of Scientific Research and Development No.865, Part IV (1942) as cited in Davies, C. N., Air Filtration, Academic Press, London, England, pp. 35-36 (1973), the theoretical expression shown in Equation 8 is described. ##EQU8##
In Happel, J., "Viscous Flow Relative to Arrays of Cylinders," American Institute of Chemical Engineering Journal, 5, pp.174-177 (1959), the theoretical expression shown in Equation 9 is described. ##EQU9##
In Kuwabara, S., "The Forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in Viscous Flow at Small Reynolds Numbers," Journal of the Physical Society of Japan, 14, pp. 527-532 (1959), the theoretical expression shown in Equation 10 is described. ##EQU10##
Further, for example, in Pich, J., Theory of Aerosol Filtration by Fibrous and Membrane Filters, Academic Press, London and New York (1966), the theoretical expression shown in Equation 11 is described. ##EQU11##
As noted previously herein, flow resistivity is an important macroscopic property for the design of porous materials, e.g., particularly, flow resistivity of a fibrous material has a large influence on its acoustical behavior. Therefore, even though various flow resistivity models are available for use, improved flow resistivity models are needed for improving the prediction of acoustical properties of porous materials, particularly fibrous materials.
Various materials, such as, for example, those modeled as described generally above, including fibrous materials, may be used in acoustical systems including multiple components. For example, an acoustical system may include a fibrous material and a resistive scrim having an air cavity therebetween. Systems and methods are available for determining various acoustical properties of materials, e.g., porous materials, and of acoustical properties of acoustical systems (e.g., acoustical properties such as sound absorption coefficients, impedance, etc.). For example, systems for creating graphs representative of absorption characteristics versus at least thickness for an absorber consisting of a rigid resistive sheet backed by an air layer have been described. This and several other similar programs are described in Ingard, K. U., "Notes on Sound Absorption Technology," Version 94-02, published and distributed by Noise Control Foundation, Poughkeepsie, N.Y. (1994).
However, although acoustical properties have been determined in such a manner, such determination has been performed with the use of macroscopic properties of materials. For example, such characteristics have been generated using macroscopic property inputs to a specifically defined program for a prespecified acoustical system for generating predesignated outputs. Such macroscopic properties used as inputs to the system include flow resistivity, bulk density, etc. Such systems or programs do not allow a user to predict and optimize acoustical properties using parameters of the materials, such as, for example, fiber size of fibers in fibrous materials, fiber shape, etc. which are directly controllable in the manufacturing process for such fibrous materials.
As indicated above, various methods by way of numerous models are available for predicting acoustical properties. However, such methods are not adequate for predicting acoustical properties of limp fibrous materials as the frames of limp fibrous materials are neither rigid nor elastic. The rigid porous material models are simpler and more numerically robust than the elastic porous material models. However, such rigid methods are not capable of predicting the frame motion induced by external force with respect to limp frames. In elastic porous material methods, the bulk modulus can be set to zero to account for the limp frame characteristic; however, the zero bulk modulus of elasticity causes numerical instability in computations of acoustical properties for limp materials, e.g., such as instability due to the singularity of a fourth order equation. Therefore, the existing porous material prediction processes are not suitable for predicting the acoustical behavior of limp fibrous materials and there exists a need for a limp material prediction method. In addition, there exists a need for methods for predicting and optimizing acoustical properties for use in the design of homogeneous porous materials and/or multiple component acoustical systems using parameters that are directly controllable in the manufacturing process of the materials.